Overview. Digital Systems, Computers, and Beyond Information Representation Number Systems [binary, octal and hexadecimal] Arithmetic Operations Base Conversion Decimal Codes [BCD (binary coded decimal)] Alphanumeric Codes Parity Bit Gray Codes. Digital Logic.
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Discrete
Discrete
Information
Inputs
Discrete
Processing
Outputs
System
System State
Memory
Control
Datapath
CPU
unit
Inputs: keyboard, mouse, wireless, microphone
Outputs: LCD screen, wireless, speakers
Input/Output
Synchronous or Asynchronous?
Time
Continuous in value & time
Analog
Digital
Discrete in value & continuous in time
Asynchronous
Discrete in value & time
Synchronous
Threshold Region
Magnetic Field Direction
Surface Pits/Light
Electrical Charge
j =  1
i = n  1
å
å
+
i
j
(Number)r
=
A
r
A
r
i
j
i = 0
j =  m
(Integer Portion)
(Fraction Portion)
+
(
)
(
)
Carries 00
Augend 01100 10110
Addend +10001+10111
Sum
Z
1
0
1
0
1
0
0
1
Single Bit Binary Subtraction with BorrowX
X
0
0
0
0
1
1
1
1
BS
BS
11
0 0
1 0
1 1
0 1
0 0
1 1
0 0
 Y
 Y
0
0
1
1
0
0
1
1
Borrows00
Minuend 10110 10110
Subtrahend 10010 10011
Difference
Exponent
Value
Exponent
Value
0
1
11
2,048
1
2
12
4,096
2
4
13
8,192
3
8
14
16,384
4
16
15
32,768
5
32
16
65,536
6
64
17
131,072
7
128
18
262,144
8
256
19
524,288
9
512
20
1,048,576
10
1024
21
2,097,152
Name
Radix
Digits
Binary
2
0,1
Octal
8
0,1,2,3,4,5,6,7
Decimal
10
0,1,2,3,4,5,6,7,8,9
Hexadecimal
16
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
integers) in hexadecimal represent:
Decimal
Binary
Octal
Hexa
decimal
(Base 10)
(Base 2)
(Base 8)
(Base 16)
00
00000
00
00
01
00001
01
01
02
00010
02
02
03
00011
03
03
04
00100
04
04
05
00101
05
05
06
00110
06
06
07
00111
07
07
08
01000
10
08
09
01001
11
09
10
01010
12
0A
11
0101
1
13
0B
12
01100
14
0C
13
01101
15
0D
14
01110
16
0E
15
01111
17
0F
16
10000
20
10
1) Convert the Integer Part
2) Convert the Fraction Part
3) Join the two results with a radix point
Repeatedly divide the number by the new radix and save the remainders. The digits for the new radix are the remainders in reverse order of their computation. If the new radix is > 10, then convert all remainders > 10 to digits A, B, …
Repeatedly multiply the fraction by the new radix and save the integer digits that result. The digits for the new radix are the integer digits in order of their computation.If the new radix is > 10, then convert all integers > 10 to digits A, B, …
1011102 = 1·32 + 0·16 +1·8 +1·4 + 1·2 +0·1
= 32 + 8 + 4 + 2
= 46
0.10112 = 1/2 + 1/8 + 1/16
= 0.5000 + 0.1250 + 0.0625
= 0.6875
6 3 5 . 1 7 7 8
Binary Number
000
001
010
011
101
110
111
Color
Red
Orange
Yellow
Green
Blue
Indigo
Violet
2n³M> 2(n – 1)
n = log2M where x , called the ceilingfunction, is the integer greater than or equal to x.
931 is coded as 1001 0011 0001
(Refer to Table 1
4 in the text)
ASCII has some interesting properties:
to 3916
.

Z span 4116
to 5A16
.

z span 6116
to 7A16
.
occurs by
flipping bit 6.
a carryover from when
punched paper tape was used to store messages.
Even Parity
Odd Parity
Message
Parity
Parity
Message


000
000


001
001


010
010


011
011


100
100


101
101


110
110


111
111


Decimal
BCD
Gray
0
0000
0000
1
0001
0100
2
0010
0101
3
0011
0111
4
0100
0110
5
0101
0010
6
0110
0011
7
0111
0001
8
1000
1001
9
1001
1000
111
000
100
000
B
0
B
1
101
001
001
110
B
2
G
2
G
1
G
111
0
010
011
101
100
011
110
010
(b) Gray Code for Positions 0 through 7
(a) Binary Code for Positions 0 through 7